Analysis of Clocked Sequential Circuits

Analysis of Clocked Sequential Circuits

AU: Dec.-03,06,07,15,19, May-07,10,11,15,19

• The behaviour of a sequential network is determined from the inputs, the outputs, a and the states of its flip-flops.

• The analysis of sequential circuit consists of obtaining a table or a diagram for the time sequence of inputs, outputs and internal

states.

Consider the sequential circuit to be analysed as shown in Fig. 3.3.1.

Let us see the steps to analyze the given synchronous sequential circuit.

1. Determine the flip-flop input equations and the output equations from the sequential circuit.

Z = α Q

T = α Q + ᾱ 

2. Derive the transition equation.

The transition equation for T flip-flop is

3. Plot the next step map for each flip-flop

4. Plot the transition table

Note If circuit consists of more than one flip-flop we have to combine the map of flip-flop to derive the transition table. (Refer next example)

5. Draw the state table

The transition table shown in Table 3.3.1 can be converted into state table as sown in Table 3.3.2. Here, new symbols to binary codes are assigned. They area = 0, b = 1.

Examples for Understanding

Example 3.3.1 Construct the transition table, state table and state diagram for the Moore sequential circuit shown in Fig. 3.3.4 (a):

Solution: 

1. Determine the flip-flop input equations and the output equations from the sequential circuit.

2. Derive the transition equations.

The transition equations for JK flip-flops can be derived from the characteristic equation of JK flip-flop as follows:

We know that for JK flip-flop

3.Plot a next-state maps for each flip-flop.

The next-state maps are:

4.Plot the transition table.

The transition table can be formed by combining the above two maps. The Table 3.3.3 shows the transition table.

Note For Moore sequential circuit output only depends on present state and not on the input.

5. Draw the state table

By assigning a = 0 0, b = 01, c = 10 and d = 11 we can write state table from the transition table as shown.

6. Draw state diagram

From the state table we can draw state diagram as shown in Fig. 3.3.5

Note In case of Moore model, the directed lines are labelled with only one binary number representing the state of the input that causes the state transition. The output state is indicated within the circle, below the present state because output state depends only on present state and not on the input.

Example 3.3.2 A sequential circuit with 2D FFs A and B and input X and output Y is specified by the following next state and output equations.

A (t+1)=AX+BX

B(t+1) = A'X

Y = (A+B) X'

i) Draw the logic diagram of the circuit.

ii) Derive the state table.

iii) Derive the state diagram.AU :Dec-06,15, May-07, Marks 16

Solution :

i)Logic diagram

ii) State table

Step 1:Plot the next-state map for each flip-flop.

Step 2:Plot the transition table.

Step 3:Draw the state table.

By assigning a = 00, b = 01, c = 10 and d=11 we can write state table from the transition table.

Example 3.3.3: A sequential circuit with two D flip-flops A and B, one input x, and one output z is specified by the following next state and output equations:

A(t + 1) = A'+B, B(t + 1) = B'x, z = A + B',

1) Draw the logic diagram of the circuit

2) Derive the state table

3) Draw the state diagram of the circuit.  AU May-15, Marks 4+3+3

Solution :

1. Logic Diagram

2. State Table

Step 1:Plot the next-state map for each flip-flop.

Step 2:Plot the transition table

Step 3:Draw the state table

By assigning a = 00, b = 01, c = 10 and d =11we can write state table from thetransition table.

3. State diagram

Example 3.3.4 A sequential circuit has four FFs ABCD and an input X is described by the following state equations.

B (t + 1) = A

C (t + 1) = B

D (t + 1) = C

i) Obtain the sequence of states when X = 1, starting from state ABCD = 0001.

ii) Obtain the sequence of states when X = 0 starting from state ABCD 0000.  AU: Dec.-06, May-07, Marks 16

Solution:

i) X = 1 and Initially ABCD = 0001.

A B C D E<= 1000,0100, 0010, 1001, 1100, 0110, 1011, 0101, 1010, 1101,1110,

1111, 0111, 0011, 0001

ii) X = 0 and Initially ABCD = 0000

A B C D <= 1000, 1100, 1110, 0111, 1011, 1101, 0110, 0011, 1001, 0100, 1010, 0101, 0010, 0001, 0000

Example 3.3.5 A sequential circuit has one flip-flop Q, two inputs x and y, and one output S. It consists of a full adder circuit connected to a D flip-flop, as shown below. Derive the state table and state diagram of the sequential circuit.   AU Dec.-07, Marks 12

Solution :

Step 1:Derive the state table

Example 3.3.6 For the circuit shown in Fig. 3.3.11, write down the state table and draw the state diagram and analyze the operation. AU: May-10, Marks 16

Solution :

Step 1:Inputs for each flip-flop

J₁= 1                    K₁ = 1

J₂= Q₁ Q₄              K₂ = Q₁

J₃= Q₂ Q₁              K₃= Q₂ Q₁

J₄= Q₁ Q₂ Q₃       K₃ = Q₁

Step 2: Transition equations for each flip-flop

For JK flip-flop

Step 3:Next state maps for each flip-flop

Step 4: Transition table

Step 5: State assignment

a = 0000,

b=0001,

C = 0010,

d = 0011,

e = 0100,

f = 0101,

g= 0110,

h = 0111,

i = 1000,

j = 1001,

k = 1010,

1 = 1011,

m = 1100,

n = 1101,

0 = 1110,

p = 1111

Step 6: State table

Step 7:State diagram

Example 3.3.7 A synchronous counter with four JK flip-flops has the following connections:

J_A = K_A = 1, J_B = Q_A D, K_B = Q_A

J_C = K_C = Q_A Q_B

J_D = Q_A Q_B Q_C and K_D Q_A

Determine the modulus n of the counter and draw the output waveforms of the

same.  AU May-11, Marks 16

           

Counter goes through states: 0 - 8 - 4 -12 - 2 -10 - 6 – 14 – 1 – 9 – 0

Thus is a modulus 10 counter.

Example 3.3.8 A sequential circuit with two D flip-flops A and B, two inputs X and Y and one output z is specified by the following input equations:

A(t + 1) = x'y + xA, B(t + 1) = x'B + xA, z = B

Draw the logic diagram of the circuit. Derive the state table and state diagram and state whether it is a Mealy or a Moore machine. AU May-19, Marks 8

Solution: 

Logic diagram :

State table :

State assignment: 00→a, 01 → b, 10 → c and 11 → d

State diagram:

For this sequential circuit output is a function of present state only and hence it is a Moore machine.

Example 3.3.9 A sequential circuit with two D flip-flops A and B, two inputs x and y; and one output z is specified by the following next-state and output equations: A(t + 1) = xy' + xB, B(t + 1) = xA + xB', z = A

i) Draw the logic diagram of the circuit. AU: Dec.-19, Marks 8

ii) List the state table for the sequential circuit  Marks 4

iii) Draw the corresponding state diagram  Marks 3

Solution: 

Logic diagram :

State table :

State assignment: 00→a, 01 → b, 10 → c, 11 → d

State diagram:

Examples for Practice

Example 3.3.10: Construct the transition table, state table and state diagram for the Mealy sequential circuit given in Fig. 3.3.17.

Example 3.3.11: Derive the transition table, state table and state diagram for Moore sequential circuit shown in Fig. 3.3.18.

Review Question

1. Explain the various steps in the analysis of synchronous sequential circuits with suitable example.  AU: Dec.-03, Marks 8